Ablowitz M.A. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons

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11.2 Power-energy-saturation equation 325

0 200 400 600 800 1000

g = 0.1

g = 0.3

g = 1.0

g = 0.1

g = 0.3

g = 1.0

Figure 11.7 Evolution of the peak of a unit Gaussian using the DM-PES

equation with s = 1 and gain parameters g = 0.1 (damped evolution), 0.3and

1.0. In the inset, the early evolution of the peaks is shown.

−10 0 10

s = 0.1

g = 0.3

μ = 17.10

g = 0.7

μ = 44.0

μ = 1.55

g = 1.0

−10 0 10

s = 1.0

μ =72

μ =33

g = 1.0

μ = 0.66

g = 0.3

g = 0.7

−50 0 50

0.5

1.5

2.5

s = 10

μ =3.93

g =1.0

μ = 1.24

g =0.7

μ =0.13

g =0.3

Figure 11.8 DM-PES solitons for various values of gain, g, and propagation

constant, μ, with map strength s = 1.

The ﬁnal mode-locking state corresponds to a soliton that can be obtained

from spectral renormalization methods (discussed in Chapter 10). A soliton

satisﬁes a ﬁxed point integral equation obtained after substituting

(Z,ω) →

(ω)e

iμZ

into the DM-PES equation (11.3). In Figure 11.8 are some typical

solitons corresponding to diﬀerent values of gain g and propagation constant

μ. As in the constant dispersive case, DM-PES solitons correspond to spe-

ciﬁc values of μ, which is the soliton analog of mode-locking. This is also

in contrast to the unperturbed case where solutions exist for all μ>0. We

see that as map strength increases the solitons become smaller in amplitude

but much more broad. As μ increases the modes become sharper and taller.

326 Mode-locked lasers

−10

1.5

0.5

−5 0 5 10

DM-PES

DMNLS

s =1.0

g =0.3

μ =0.66

Figure 11.9 Typical comparison between the solutions of the DM-PES

equation (11.3) and the pure DMNLS equation for s = 1andg = 0.3.

Each mode-locked state is aﬀected by strong saturation eﬀects. Hence the

soliton is essentially a solution of the unperturbed DM-PES equation (10.25)

(where g(ζ) is replaced by n(ζ) in the notation of this section); i.e., this is

the DMNLS equation that we obtain when we omit the gain and damping

terms. This is indicated in Figure 11.9 for typical parameters. The main eﬀect

of the gain-damping is to isolate the parameters for which mode-locking

occurs.

11.2.3 Remarks on perturbation theory

We have studied the behavior of both single solitons and soliton strings in

ML lasers in detail via perturbation theory (Ablowitz et al., 2009a). Here we

brieﬂy mention some of the key results in the case of constant dispersion,

taking d

= 1. Since the pulses have been found to be well approximated by

solutions of classical NLS equation, we assume a soliton solution of the NLS

form written as

u = ue

iφ

, u = η sech(ηθ),ξ=



Vdz+ t

(11.4a)

θ = t − ξ, σ =





μ +



dz + σ

,φ= Vθ + σ, (11.4b)

with parameters for the height (η, μ = η

/2), velocity (V), temporal shift (t

)

and phase shift (σ

) assumed to vary adiabatically, i.e., slowly, in z.

We consider the eﬀects of gain, ﬁltering and loss as perturbations of the NLS

equation, taking d

= n = 1 in the PES equation,

u = R[u] (11.5)

11.2 Power-energy-saturation equation 327

where the right-hand side F contains the gain-damping terms and is assumed

small. Recall

R =

1 + E/E

τu

1 + E/E

−

1 + P/P

and E =



∞

−∞

|u|

dt, P = |u|

. While the individual contributions of terms in F

may not always be small, the combined eﬀects are small; as discussed above

they provide the mode-locking mechanism (Ablowitz and Horikis, 2008).

We ﬁnd the equations for the evolution of the parameters η, V, t

, σ

requiring the leading-order solution, (11.4), to satisfy a set of integrated quan-

tities related to the (unperturbed) conservation laws suitably modiﬁed by the

terms in R[u]. These can be derived directly from (11.5)as



= 2Im



R[u]u

∗

(11.6a)



∗

= 2Re



R[u]u

∗

(11.6b)



= −Im



∗

+ 2Im



tR[u]u

∗

(11.6c)



∗

= −



∗

tμ

+ Re



∗

− Re



R[u]u

∗

, (11.6d)

where all integrals are taken over −∞ < t < ∞.

When (11.4) are substituted into (11.6) the modiﬁed integral equations

reduce to a system of diﬀerential equations that govern the evolution and

mode-locking of soliton pulses:

dη

= g



2η

2η + 1



− τ

2η + 1



+ 2V



+ l



2η

a − b

log







(11.7a)

= −τV



2η + 1



(11.7b)

= 0 (11.7c)

dσ

= 0; (11.7d)

here a and b are the roots of the polynomial x

+ 2(1 + 2η

)x + 1 = 0 (we can

choose a and b to be either root). Note, the ﬁrst two equations are independent

of t

and σ

, both of which are constant. We only consider parameters in the

domain η ≥ 0 (since η<0 can be represented as a positive η with a π phase

shift).

328 Mode-locked lasers

Consider the case V = 0; this is the only equilibrium for V and it is stable

for η>0. In this case we ﬁnd an equation for η independent of any other

parameters

dη

= g



2η

2η + 1



− τ

2η + 1





+ l



2η

a − b

log







which accurately predicts the mode-locking behavior found numerically.

Clearly, η has at least one equilibrium at 0, which can be proved to be stable for

g < l and unstable for g > l.Forg > l there is a second equilibrium that is stable

and corresponds to the mode-locked solution, as indicated in Figure 11.10.

Weakly interacting solitons can also be studied analytically by considering

the interaction terms as a small perturbation (Karpman and Solov’ev, 1981).

We refer the interested reader to Ablowitz et al. (2009a) for further details.

0 2 4 6

−0.5

0.5

g = 0.3

g = 0.5

g = 0.7

g =0.9

0 20 40 60 80 100

|u(z,0)|

Numerical

Asyptotics

g = 0.5

g = 0.25

g = 0.05

Figure 11.10 Phase portrait of the amplitude equation for several values

of g > l illustrating the single stable equilibrium (top). Comparison of

asymptotic and numerical results for mode-locking pulses (bottom).

11.2 Power-energy-saturation equation 329

11.2.4 The normal regime

The dynamics of pulses evolving under the PES equation in the normal regime

< 0) are studied next. As above, all terms are kept constant and only the

gain parameter g is changed. Typical values are taken: d

= −1 (i.e., the normal

regime), n = 1, E

= P

= 1,τ = l = 0.1. We study the evolution of an initial

unit Gaussian pulse peak for diﬀerent values of the gain parameter g; see Fig-

ure 11.11. When g = 0.1 the pulse decays quickly due to excessive loss with

no noticeable oscillatory or chaotic behavior; the pulse exhibits damped evolu-

tion. When g = 0.5, due to the loss in the system the pulse undergoes an initial

decrease and then very slowly dissipates. When g = 1.5 a localized evolution

is obtained. With suﬃcient gain in the system, the pulse amplitude initially

0 100 200 300 400 500

0.2

0.4

0.6

0.8

|u(z,0)|

g = 0.1

g = 0.5

g = 1.5

−500

−250

250

500

200

400

0.5

|u|

−500 0 500250−250

0.1

0.2

0.3

0.4

|u(400,t)|

−500 0 500

400

phase

Figure 11.11 Evolution of the pulse peak of an initial proﬁle under the PES

equation, in the normal regime (d

= −1) with diﬀerent values of gain. Top:

the damped pulse-peak evolution is shown with a dashed line. Middle: the

complete evolution is given for g = 1.5. Bottom: the resulting soliton and the

corresponding phase in the inset.

330 Mode-locked lasers

decreases but then a steady state is reached. The sharp and narrow initial Gaus-

sian rapidly evolves into a wide but ﬁnite-energy pulse. Three regimes are

observed: (a) when the loss is much greater than the gain the pulse decays

to zero; (b) when the loss is again a prominent eﬀect but suﬃcient gain exists

in the system to sustain a very slowly decaying evolution resulting in a “quasi-

soliton” state; and (c) the soliton regime above a certain value of gain. When

a localized state results in the normal regime, there is a strong phase eﬀect,

as indicated in Figure 11.11. This is unlike the anomalous regime where

the pulse’s temporal phase is constant and the pulses have relatively much

narrower width.

As indicated above, the values for the gain, loss and ﬁltering parameters

provide operating regions of the laser system that depend critically on the gain

parameter. The parameters d

, E

, P

, g, τ, l were chosen to have typical values.

Changing these values to more closely correspond to experimental data shows

that the above observations are quite general (see Ablowitz and Horikis, 2008,

2009b).

Bi-solitons

As mentioned above, individual pulses of (11.2) in the normal regime exhibit

strong chirp and cannot be identiﬁed as the solutions of the “unperturbed”

NLS equation. Indeed, the classical NLS equation does not exhibit decaying

“bright” soliton solutions in the normal regime. If we begin with an initial

Gaussian, u(0, t) = exp(−t

), the evolution of the PES equation mode-locks

into a fundamental soliton state. On the other hand, we can obtain a higher-

order antisymmetric soliton, i.e., an antisymmetric bi-soliton, one that has its

peak amplitudes diﬀering in phase by π. Such a state can be obtained if we start

with an initial state of the form u(0, t) = t exp(−t

) (i.e., a Gauss–Hermite poly-

nomial). The evolution results in an antisymmetric bi-soliton and is shown in

Figure 11.12 along with a comparison to the proﬁle of the single soliton. This

is a true bound state. Furthermore, the results of our study ﬁnding antisymmet-

ric solitons in the normal regime are consistent with experimental observations

(Chong et al., 2008a).

It is interesting that the normal regime also exhibits higher soliton states

when two general initial pulses (e.g., unit Gaussians) are taken suﬃciently far

apart. The resulting pulses, shown in Figure 11.13, individually have a similar

shape to the single soliton of the normal regime with lower individual energies.

These pulses if initially far enough apart can have independent chirps that may

be out of phase by an arbitrary constant. We again ﬁnd that the minimum dis-

tance required for the two solitons to remain apart is d

∗

≈ 9α which is a good

estimate for the required pulse separation just as in the constant anomalous

dispersive case! These pulses are “eﬀective bound states” in that after a long

11.2 Power-energy-saturation equation 331

−200 0 200 400−400

200

400

600

800

1000

0.1

0.2

0.3

0.4

0.5

0.6

−400 −200 0 200 400

0.1

0.2

0.3

0.4

|u(1000,t)|

−400 0 400

200

phase

−50 0 50

155

165

phase

single soliton

bi−soliton

Figure 11.12 Evolution of the antisymmetric soliton (top) and the antisym-

metric bi-soliton, in the normally dispersive regime, superimposed with the

corresponding single soliton (bottom) at z = 1000. Here g = 1.5.

−2000 −1000 0 1000 2000

0.05

0.1

0.15

0.2

|u(1000,t)|

−2000 0 2000

400

phase

Figure 11.13 Symmetric two-soliton state of the normal regime with in-phase

pulses. The phase structure is depicted in the inset. Here g = 1.5.

332 Mode-locked lasers

distance they separate very slowly. Additional higher-order 3-, 4-, ... soliton

states can also be found. We refer the reader to Ablowitz et al. (2009c)for

further details.

Asymptotic approximations of normal solitons

Using the spectral renormalization method, we have found localized pulses in

the normal dispersive regime for a broad range of parameters, provided that

suﬃcient gain is present in the system. These pulses are slowly varying in t,

with a large phase. This suggests that by assuming a slow time-scale in the

equation and using perturbation theory (i.e., a WKB-type expansion) the soli-

ton system can be reduced to simpler ordinary diﬀerential equations for the

amplitude and the phase of the pulse. The ﬁrst step before performing perturba-

tion theory is to write the solution of (11.2) in the form u(z, t) = R exp(iμz+iθ),

where R = R(t) and θ = θ(t) are the pulse amplitude and phase, respectively.

Substituting into the equation and equating real and imaginary parts and calling

 = 1/E

,δ= 1/P

, we get

−μR −



− Rθ



+ R

= −

1 + E

(2R

+ Rθ

) (11.8a)

−

(2R

+ Rθ

) =

1 + E

R +

1 + E

−

1 + δR

R. (11.8b)

We then take the characteristic time length of the pulse to be such that we can

deﬁne a scaling in the independent variable of the form ν = T/t or T = νt

so that R = R(νt), θ

= O(1) (i.e., θ is large) and θ

= O(ν) where ν  1.

Then (11.8a) becomes

−μR +

Rθ

+ R

−

1 + E

(2νR

+ Rθ

The leading-order equation is

(μ − R

). (11.9)

Using the same argument and this newly derived equation (11.9) we then ﬁnd

that (11.8b) to leading order reads

= − sgn(t)



2(μ − R

)/d

− 2μ



1 + E

−

4τ(μ − R

)/d

1 + E

−

1 + δR



. (11.10)

This is now a non-local ﬁrst-order diﬀerential equation for R = R(t), since

E =



∞

−∞

dt. From the above equation (11.9), imposing θ

(t = 0) = 0, it

follows that μ ≈ R

(0).

To remove the non-locality another condition is needed and it is based on the

singular points of (11.10). Recall that R(t) is a decaying function in t ∈[0, +∞)

11.2 Power-energy-saturation equation 333

−400 −200 0 200 400

0.05

0.1

0.15

0.2

0.25

0.3

R(t)

PES equation

reduced system

parabolic profile

−500 0 500

200

θ(t)

Figure 11.14 Solutions of the complete PES equation and the reduced sys-

tem, in the normally dispersive regime. The relative parabolic proﬁle is also

shown. Here g = 1.5.

and μ ≈ R

(0) ≥ R(t)

. Thus there exists a point in t such that the denominator

in the equation becomes zero. To remove the singularity we require that the

numerator of the equation is also zero at the same point; that leads to

1 + E =



g −

2τ



1 +

2δ



Thus (11.10) is now a ﬁrst-order equation that can be solved by standard

numerical methods and analyzed by phase plane methods. The resulting solu-

tions from the PES equation and the reduced equations are compared in

Figure 11.14. Notice that here μ = 0.1216 while R

(0) = 0.1218.

References

Ablowitz, M.J., Bakirtas, I., and Ilan, B. 2005. Wave collapse in nonlocal nonlinear

Schr

odinger systems. Physica D, 207, 230–253.

Ablowitz, M.J., and Biondini, G. 1998. Multiscale pulse dynamics in communication

systems with strong dispersion-management. Opt. Lett., 23, 1668–1670.

Ablowitz, M.J., Biondini, G., Biswas, A., Docherty, A., Hirooka, T., and Chakravarty,

S. 2002a. Collision-induced timing shifts in dispersion-managed soliton systems.

Opt. Lett., 27, 318–320.

Ablowitz, M.J., Biondini, G., and Blair, S. 1997. Multi-dimensional pulse propagation

in non-resonant χ

(2)

materials. Phys. Lett. A, 236, 520–524.

Ablowitz, M.J., Biondini, G., and Blair, S. 2001a. Nonlinear Schr

odinger equations

with mean terms in non-resonant multi-dimensional quadratic materials. Phys.

Rev. E, 63, 605–620.

Ablowitz, M.J., Biondini, G., Chakravarty, S., and Horne, R.L. 2003a. Four wave

mixing in dispersion-managed return-to-zero systems. J. Opt. Soc. Am. B, 20,

831–845.

Ablowitz, M.J., Biondini, G., Chakravarty, S., Jenkins, R.B., and Sauer, J.R. 1996.

Four-wave mixing in wavelength-division-multiplexed soliton systems: damping

and ampliﬁcation. Opt. Lett., 21, 1646–1648.

Ablowitz, M.J., Biondini, G., and Olson, E. 2000b. On the evolution and interaction

of dispersion-managed solitons. In: Massive WDM and TDM Soliton Trans-

mission Systems, edited by Akira Hasegawa. Kyoto, Japan: Kluwer Academic,

pp. 362–367.

Ablowitz, M.J., and Clarkson, P.A. 1991. Solitons, Nonlinear Evolution Equations and

Inverse Scattering. Cambridge: Cambridge University Press.

Ablowitz, M.J., Docherty, A., and Hirooka, T. 2003b. Incomplete collisions in

strongly dispersion-managed return-to-zero communication systems. Opt. Lett.,

28, 1191–1193.

Ablowitz, M.J., and Fokas, A.S. 2003. Complex Variables: Introduction and

Applications. Second edition. Cambridge: Cambridge University Press.

Ablowitz, M.J., Fokas, A.S., and Musslimani, Z. 2006. On a new nonlocal formulation

of water waves. J. Fluid Mech., 562, 313–344.

Ablowitz, M.J., and Haberman, R. 1975. Resonantly coupled nonlinear evolution

equations. Phys. Rev. Lett., 38, 1185–1188.

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